UNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK

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1 UNIVERSITY OF CAMBRIDGE Faculty of Mathematics MATHEMATICS WORKBOOK August, 07

2 Introduction The Mathematical Tripos is designed to be accessible to students who are familiar with the the core A-level syllabus (i.e. modules C C4) and a typical FP syllabus. If you have not followed an A- level course, you can see what is in these syllabuses by looking, for example, at and hunting for qualifications and then when you find Mathematics and Further Mathematics A, hunt for specifications. The purpose of this workbook is to present a set of material that would be useful to know before starting the Tripos. It does not, of course, contain everything you should know. Rather than present a list of topics, the workbook contains questions on each topic that are supposed to be straightforward; by tackling these questions, you will see how much knowledge is expected. Most of you will be familiar with most of the material covered here. It is still worth sketching out a solution to each question: it will be good revision and anyway you never really know that you have understood a problem until you do it. The answers are given at the end of the workbook. Note that none of the questions requires the use of a calculator. If you find some of the material is unfamiliar, you should look it up. You will find most of it in any standard A-level (or the equivalent) text; and you may well find a helpful internet source just by googling the topic. If a particular area is really unfamiliar to you, it would be worth doing exercises from a text book to supplement those given here. If you have difficulties with some questions, don t worry; you will have opportunities to cover the material when you get to Cambridge. You may also find that you need to get into the swing of mathematics again after your long break (especially if you took a gap year). The best way to do this is to practice problem-solving. One source of problems is past STEP papers (available on the Cambridge Assessment STEP web site, which will be especially useful if you did not take STEP. If you have the STEP papers of this summer, you might like to have another go at them to get back up to speed. Please comments or corrections to undergrad-office@maths.cam.ac.uk. August, 07

3 Algebra Although computers and even calculators are very good at algebra, all mathematicians agree that it is important to be able to do routine algebra quickly and accurately. You should be able to state elementary series expansions including binomial, sine and cosine, and ln series. Factorization. Factorize the following polynomials: x 3x + ; 3x 3 3x 6x ; x x ; (iv) x 3 ; (v) x 4 3x 3 3x + x 6. Notes In part you will need the quadratic formula to find the factors ; part (iv) has one linear and one quadratic factor (or three linear factors two of which are complex) ; for part (v) you can use the factor theorem. Inequalities. Find the values of x for which x 3 < x + 3x. 3 Partial fractions. Express the following in partial fractions: (x + )(x ) ; x 3 + ; 4x + (x + ) (x ) ; (iv) x 7 (x )(x + ). Note It is best for these purposes not to use the cover-up rule ; there are at least two other ways which involve elementary mathematics, whereas the cover-up rule works for more sophisticated reasons and to most users is simply a recipe (which does not always work). 4 Completing the square. Find the smallest value (for real x and y) of: x x + 6 ; x 4 + x + y 4 y + 3 ; sin x + 4 sin x. Note Of course, you could find the smallest value by calculus, but expressing the function as a perfect square plus a remainder term is a surprisingly useful technique for example, when integrating a function with a quadratic denominator.

4 5 Exponentials and logs. The exponential function e x can be defined by the series expansion e x = + x + x! + x3 3! +. Use this definition to show that dex dx = ex. The natural log function ln t can be defined (for t > 0) as the inverse of the exponential function, so that ln e x = x. Set t = e x and use the relationship dx / dt dt = to show that dx d ln t dt Assuming that the exponential function has the property e s e t = e s+t, prove that ln(xy) = ln x+ln y. = t. (iv) The definition of a x for any a is e x ln a. Prove that a x a y = a x+y and a x b x = (ab) x. Note If you have not thought of defining a x in this way, it is worth considering how else you could give it a meaning when x is not an integer. 6 Binomial expansions. Find the coefficient of x k (for 0 k 0) in the binomial expansion of ( + 3x) 0. Use the binomial theorem to find the expansion in powers of x up to x 4 of ( + x + x ) 6, by writing it in the form ( + (x + x ) ) 6. Use binomial expansions to find the expansion in powers of x up to x 4 of ( x 3 ) 6 ( x) 6. (iv) Find the first four terms in the binomial expansion of ( + x). 7 Series expansions of elementary functions. Using only the series expansions sin x = x x 3 /3! + x 5 /5! +, cos x = x /! + x 4 /4! +, e x = + x + x /! + x 3 /3! + and ln( + x) = x x / + x 3 /3 +, find the series expansions of the following functions: tan x (up to the x 5 term) ; sin x cos x (up to the x 5 term) ; (v) e x + e x cos x x (up to the x 5 term) ; (iv) ln(e x ) (up to the x 3 term) ; (up to the x term). Notes Do part without using a trig. formula, and compare your answer with the expansion for sin(x). The function in part is cosh x, of which more later. The interesting thing about the function in part (v) is that the series shows it is well-behaved in the limit x = 0, despite appearances. 3

5 8 Proof by induction. Prove by induction that the following results are valid. r a + ar + ar + + ar n n = a. r n r 3 = 4 n (n + ). r= Note Methods of proof will be discussed in some detail in our first term course Numbers and Sets. Most students will have met mathematical induction at school ; if you haven t, you will probably want to try out this straightforward but important method of proof. 9 Arithmetic and geometric progressions. Find the sum of all the odd integers from to 99. Evaluate Find sin θ + sin 3 θ + 4 sin 5 θ +. (What ranges of values of θ are allowed?) (iv) Estimate roughly the approximate number of times a piece of paper has to be torn in half, placing the results of each tearing in a stack and then doing the next tearing, for the stack of paper to reach the moon. Note The dots in and indicate that the series has an infinite number of terms. For part, recall that the expansion a(+r+r + ) only converges if < r <. You may find the approximation 0 3 = 0 useful for part (iv). The distance from the Earth to the Moon is about km. Trigonometry It is not necessary to learn all the various trigonometrical formulae; but you should certainly know what is available. The double-angle formulae, such as tan x = tan x/( tan x) are worth knowing, as are the basic formulae for sin(a ± B), cos(a ± B) and tan(a ± B), the Pythagoras-type identities, such as sec x = + tan x, and a few special values, such as sin π/4 = /, that can be deduced from right-angled triangles with sides (,, ) or (, 3, ). 0 Basic identities. Starting from the identity sin(a + B) = sin A cos B + cos A sin B, use the basic properties of the trig. functions (such as sin( A) = sin A) to prove the following: sin(a B) = sin A cos B cos A sin B ; cos(a + B) = cos A cos B sin A sin B ; tan(a + B) = tan A + tan B tan A tan B ; (iv) sin C + sin D = sin (C + D) cos (C D) ; a + b (v) tan a + tan b = tan + nπ. ab Notes For part, recall that cos A = sin(π/ A). You can use part to help with part (v). What exactly does part (v) mean? 4

6 Trig. equations. Solve the following equations: (a) sin(x + π 6 ) + sin(x π 6 ) = 3 ; (b) cos x + cos x + cos 3x = 0. Write down the value of cot(π/6) and use a double angle formula to show that cot(π/) satisfies the equation c 3c = 0. Deduce that cot(π/) = + 3. Trig. identities using Pythagoras. Prove the following identities: tan θ + cot θ = sec θ cosec θ ; (cot θ + cosec θ) = + cos θ cos θ ; cos θ = t t t ; sin θ = ; tan θ = + t + t t, where t = tan θ. Complex numbers A complex number z can be written as x + iy, where x is the real part, y is the imaginary part and i =. The modulus of z (written z or r) is (x + y ) and the argument (written arg z or θ) is defined by x = r cos θ, y = r sin θ and π < θ π. The complex conjugate of z (written z ) is x iy. The inverse, z, of z is the complex number that satisfies z z = (for z 0). 3 Algebra of complex numbers. Use the definitions above with z = x + iy and z = x + iy to show: z z = (x x y y ) + i(x y + y x ) ; z = zz ; z = z z ; (iv) z z = z z ; (v) arg(z z ) = arg z + arg z (assume that 0 < arg z π/4 and 0 < arg z π/4). Give a sketch of the x-y plane (called also the complex plane or the Argand diagram) showing the points representing the complex numbers z = i, z = 3 + i. Verify results, (iv) and (v) for these numbers. 4 De Moivre s theorem. De Moivre s theorem is not in the FP syllabus for most examination boards, but it is so important that you should get to know it if you haven t already met it. Show by means of series expansions that cos θ + i sin θ = e iθ and deduce that cos θ i sin θ = e iθ and that z = re iθ. Deduce also that cos θ = (eiθ + e iθ ) and sin θ = i (eiθ e iθ ). Use the above result to show that re iθ = (for real θ and r > 0) if and only if r = and θ = nπ for some integer n. Use part to find the three distinct roots of the equation z 3 =. Draw them on the complex plane and convert them from modulus-argument form to real-imaginary form. 5

7 5 Geometry of the complex plane. Show by transforming to Cartesian coordinates that the equation z c = r, where c is a complex number, describes a circle or a point. Show that the equation arg z = α, where α is a constant, describes a line segment. Show by means of a diagram that z + z z + z for any two complex numbers. Deduce that z z 3 z + z 3 and z 4 z z 4 z for any complex numbers z, z, z 3 and z 4. Under what circumstances does the equation z + z = z + z hold? Hyperbolic functions Prior knowledge of hyperbolic functions is not assumed for our mathematics course, but it is worth getting to know the definitions and basic properties, which are given below. Hyperbolic functions are very similar to trigonometric functions, and many of their properties are direct analogues of the properties of trigonometric functions. The definitions are sinh x = ex e x, cosh x = ex + e x, tanh x = ex e x e x + e x and cosech x = (sinh x), sech x = (cosh x), coth x = (tanh x). 6 Basic properties. Give a rough sketch of the graphs of the six hyperbolic functions. Show from the above definitions that cosh x sinh x = ; d(sinh x) dx = cosh x ; d(cosh x) dx = sinh x ; cosh(x + y) = cosh x cosh y + sinh x sinh y ; (iv) cosh(ix) = cos x, sinh(ix) = i sin x, cos(ix) = cosh(x), sin(ix) = i sinh x. Notes You can use (iv) to prove and using the corresponding trig. identities. In fact, (iv) is behind the rule which says that any formula involving trig. functions becomes the corresponding formula involving hyperbolic functions if you change the sign of every product of two odd functions (such as sin x or tan x). 7 Further properties. Use the definitions, and the results of the previous question, to show that sech x = tanh d(tanh x) x = sech x dx d (sinh x) dx = sinh x (iv) sinh x = ln ( x + (x + ) ). Note For part (iv), note that sinh x is the inverse function, not the reciprocal. Conic sections You should be familiar with basic coordinate geometry (lines, circles, tangents and normals to curves, etc). Conic sections (parabola, ellipse and hyperbola) are introduced in the first term course Vectors and Matrices, but you would find it very useful to do a little preliminary work on coordinate and parametric equations for conic sections if you have not seen them before. They are not only important in pure mathematics: they also arise, for example, as celestial orbits in the second term Dynamics course. 6

8 8 Basic definitions. Parabola. The point (x, y) has the property that its distance from the point (a, 0) is equal to its distance from the line x = a. Sketch the locus of the point using this information. Show that y = 4ax. Ellipse and hyperbola. The point (x, y) has the property that its distance from the point (ae, 0) is e times its distance from the line x = ae. Sketch the locus of the point for 0 < e < and for e >. Show that x /a + y /b = if 0 < e < and that x /a y /b = if e >, where b = a e. Rectangular hyperbola. Show that the hyperbola x /a y /b = becomes the rectangular hyperbola XY = in the new coordinates given by x = a(x + Y )/, y = b(x Y )/. 9 Parametric equations. Show that the curves with parametric form (x, y) = (at, at), (a cos θ, b sin θ) and (a cosh θ, b sinh θ) are conic sections. 0 Polar coordinates. By converting to cartesian coordinates, show that: r cos(θ α) = c describes a straight line ; r = l cos θ describes a circle ; r cos(θ + α) cos(θ α) = c describes a hyperbola ; (iv) r = k cos θ + m can describe any conic section, the type depending on the values of k and m. Notes Here, c, k, l, m and α are constants. Special cases, such as α = 0 in part, can be ignored. In each case, you should give a careful description (x and y intercepts, e, etc) of the curves. Matrices and vectors Matrices and vectors arise in all branches of mathematics and form an indispensable part of a mathematician s toolkit. You will have met them in FP (or the equivalent if you did not take A-levels) and should be familiar with the basic properties. It is also worth understanding the basic geometric uses of vectors. We consider here only matrices Matrix multiplication. Let A =, B = 3 ( 0 ), C = and I = 0. 0 Show that AB BA. (This shows that matrix multiplication is not commutative.) Show that (AB)C = A(BC). (This illustrates that matrix multiplication is associative). a b Without using any formulae, find a matrix D = such that AD = I. c d (iv) Show that (AB) T = B T A T. (v) Show that det AB = det A det B. Notes The symbol ( T ) denotes the transpose of the matrix, i.e. the matrix obtained by exchanging rows T a b a c a b and columns: =. The determinant of is ad bc. c d b d c d 7

9 Position vectors. Show that the points with position vectors 0,, 0 4 3, lie on a straight line and give the equation of the line in the two forms r = a + λb x x 0 c = y y 0 d = z z 0. e 3 Scalar products. The three vectors A, B and C are defined by Order the vectors by magnitude. A = (, 3, 4), B = (,, 3), C = (3, 3, ). Use the scalar product to find the angles between the pairs of vectors (a) A and B, (b) B and C, leaving your answer in the form of an inverse cosine. Find the lengths of the projections of the vectors (a) A onto B, (b) B onto A. (iv) Find an equation of the plane, in the form r = a+λb+µc, through the points with position vectors A, B and C. Show that the normal to this plane is (, 0, ) and find an equation of the plane in the form r.n = p, where n is a unit vector. Note The projection of A onto B is the component of A in the direction of the vector B. Differentiation Differentiation of standard functions, products, quotients, implicit function expressions, and functions of a function (using the chain rule) should be routine. 4 Direct differentiation. Differentiate y(x) with respect to x in the following cases: y = ln ( x + ( + x ) ) ; y = a x ; y = x x ; (iv) y = sin x + x. Notes Simplify your answer to part. For, see the definition in question 5(iv). Can you see why the answer to (iv) is surprisingly simple? 5 Parametric differentiation. Show that if x = a cos θ, y = b sin θ then d y < 0 for y > 0. dx 6 Stationary points. Find the stationary points of the function f(x) = x x + a, where a > 0, classifying them as either maximum or minimum. calculator). Sketch the curve (without using a 8

10 Integration You need to be able to recognise standard integrals (without having to leaf through a formula book) and evaluate them (referring to your formula book, if necessary). You need to be familiar with the techniques of integration by parts and by substitution. 7 Indefinite integrals. Calculate the following integrals: (v) (vii) (ix) a x dx ; e ax cos(bx) dx ; (iv) dx ; (vi) x3 sec x dx ; tan x dx ; (viii) (x) x x + 6 dx ; e ax e ibx dx ; cosec x dx ; (c + m y ) dy ; x 3 e x dx. Notes For part, see question 5(iv). For part, see question 4. Note that you can obtain from (iv) by taking the real part. Use partial fractions for part (v). For (vi) and (vii), use the substitution t = tan(x/) rather than the trick of multiplying top and bottom by e.g. sec x + tan x. Try also deriving (vii) from (vi) by means of the substitution y = π/ x. For (viii), either substitute my = c sinh θ or, if you are not keen on hyperbolic functions, see question 4. Use integration by parts for (ix) and (x). Differential equations There is a course on differential equations in the first term for which very little knowledge is assumed. 8 First order equations. Find the general solution (i.e. with a constant of integration) of the following equations. (v) y dy dx = x ; dy dx = my ; x dy dx = y + ; (iv) dy dx y tan x = ; dy dx = c k y ; (vi) z dz dy + k y = 0. Note For part (iv), find a function ( integrating factor ) w(x) such that the equation can be written d(wy) dx = w. 9

11 9 Second order equations. satisfies Show, by substituting y = e mx into the equation, that there are two values of m for which e mx where a, b and c are constants. a d y dx + b dy + cy = 0, dx ( ) Show (by substitution) that if both y and y satisfy the equation ( ), then so also does the function y defined by y = Ay + By, where A and B are constants. Find the two values of m for which e mx satisfies the equation d y dx + 3 dy dx + y = 0 and write down a solution that contains two arbitrary constants. You have just solved the above equation (y + 3y + y = 0) by guessing solutions. Instead, let z = y + y. Then show that z + z = 0, solve for z and then solve the resulting first-order differential equation of y, thereby proving that the solution you found above is in fact the most general solution. (iv) Show by substitution that e px sin qx satisfies the equation d y dy p dx dx + (p + q )y = 0. Find the general solution by the method outlined in the second paragraph of above. 30 Simple harmonic motion. Show that the equation d y dx + k y = 0 (y real) can be written as z dz dy + k y = 0, where z = dy dx. Show that z = ± (c k y ), where c is a constant of integration, (compare questions 8(v) and 8(vi)) and hence show that y = R sin k(x x 0 ), where R and x 0 are constants. (R = c/k and x 0 is a new constant of integration.) Repeat the steps of part on the equation to obtain In this case, is it the general solution? d y dx k y = 0 y = R sinh k(x x 0 ). 0

12 Answers. (x )(x ) 3x(x )(x + ) (x ( 5))(x ( + 5)) (iv) (x )(x + x + ) (v) (x 3)(x ) (x + ).. x < or 0 < x < x x + (x + ) x + + x ( 3 x + x ) x x + (iv) x + x ; ; 3 (smallest when sin x = ) k 0 k 0! k!(0 k)! Same as (iv) + 6x + x + 50x x 4 ( + 4 x 3 x + 8 x3). 7. x + 3 x3 + 5 x5 x 3 x3 + 5 x5 + x + 4 x4 (iv) x (v) 3 x sin θ cos θ ( π/4 + nπ < θ < π/4 + nπ) (iv) about 4 times.. (ia) nπ + ( ) n π/6 (ib) nπ/ + π/4, nπ ± π/3. 3. z =, arg z = π/4, z =, arg z = 5π/6. 4., e iπ/3, e iπ/3 or, ( ± i 3)/ r = 0 + λ x = y = z 3. A > C > B ; cos 7/( 9), cos 5/( 77), 7/ 4, 7/ 6 (iv) r = 3 + λ + µ 0, r. / 0 4 / = 5/. 4. x + a x ln a x x ( + ln x) (iv) + x (sin x x + = tan x).

13 6. Maximum at ( a, ) ( ), minimum at a,. a a 7. (v) ln a ax + C a + b (a cos bx + b sin bx)eax + C 6 ln x + x + (x ) + x + tan + C 3 3 (iv) (vi) ln tan(x/) + C (vii) ln tan x + C 5 5 a + ib e(a+ib)x + C + tan(x/) + C tan(x/) ( (viii) m ln (my + ) m y + c )/c + C or m sinh (my/c) + C (ix) x tan x ln( + x ) + C (x) (x )e x + C 8. y = x + C y = Ce mx y = Cx (iv) y = C sec x + tan x (v) ky = c sin k(x x 0 ) (vi) z + k y = C 9. Solution of am + bm + c = 0 m = and m = ; y = Ae x + Be x (iv) Set z = y (p + iq)y.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations: